Graph potentials and topological quantum field theories

TL;DR

This paper introduces graph potentials linked to colored trivalent graphs, establishing their topological invariance and constructing a related topological quantum field theory with an efficient computational method for its partition function.

Contribution

It defines graph potentials and demonstrates their dependence solely on the homotopy type, leading to a novel topological quantum field theory with practical computation techniques.

Findings

Graph potentials depend only on the homotopy type of the graph.

A new topological quantum field theory is constructed from these potentials.

An efficient method for computing the partition function is provided.

Abstract

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and use this to define a topological quantum field theory. A similar construction was recently introduced independently by Kontsevich--Odesskii under the name of multiplicative kernels. We end our paper by giving an efficient computational method to compute its partition function. This is the first paper in a series, and we give a survey of the applications of graph potentials in the other parts.